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Tschirnhaus transformation
Mathematical term; type of polynomial transformation / From Wikipedia, the free encyclopedia
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In mathematics, a Tschirnhaus transformation, also known as Tschirnhausen transformation, is a type of mapping on polynomials developed by Ehrenfried Walther von Tschirnhaus in 1683.[1]
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Simply, it is a method for transforming a polynomial equation of degree with some nonzero intermediate coefficients,
, such that some or all of the transformed intermediate coefficients,
, are exactly zero.
For example, finding a substitution
for a cubic equation of degree ,
such that substituting yields a new equation
such that ,
, or both.
More generally, it may be defined conveniently by means of field theory, as the transformation on minimal polynomials implied by a different choice of primitive element. This is the most general transformation of an irreducible polynomial that takes a root to some rational function applied to that root.